Relations & Functions

CISC1100, Spring 2013

Fordham Univ

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Overview: relations & functions

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Binary relations Defined as a set of ordered pairs

Graph representations

Properties of relations Reflexive, Irreflexive

Symmetric, Anti-symmetric

Transitive

Definition of function

Property of functions ◦ one-to-one

◦ onto

◦ Pigeonhole principle

Inverse function

Function composition

Relations between people

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Two people are related, if there is some family connection between them

We study more general relations between two people:

“is the same major as” is a relation defined among all college students

If Jack is the same major as Mary, we say Jack is related to Mary under “is the same major as” relation

This relation goes both way, i.e., symmetric

“is older than” defined among a set of people

This relation does not go both way

“ is facebook friend with”, …

Relations between numbers

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Comparison relation

=, <, >, <=, …

Other relations

Add up to 10, e.g., 2 and 8 is related under this relation, and so

is 5 and 5, …

Is divisible by

a is divisible by b, if after dividing a by b, we get a remainder of 0

E.g. 6 is divisible by 2, 5 is not divisible by 2, 5 is divisible by 5, …

Relation is a graph

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nodes (solid small circle): cities,…

Arcs: connecting two cities, … that are related (i.e., connected by a direct flight) with Arrows: the direction of the “relation”…

Ex: Relations between sets

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Given some sets, {},{1}, {2}, {1,2}, {1,2,3}

“Is a subset of” relation:

{} is a subset of {1}

{1} is a subset of {1,2}, …

Practice: draw the graph for each of above relations

“Has more elements than” relation:

{1} has more elements than {}, …

“Have no common elements with” relation:

{} has no common elements with {1},

{1} has no common elements with {2}…

Binary relations: definition

Relations is defined on a collection of people,

numbers, sets, …

We refer to the set (of people, numbers, …) as the domain of

the relation, denoted as S

A rule specifies the set of ordered pairs of objects in S that

are related

Rule can be specified differently

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Ways to describe the rule

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Consider domain S={1,2,3}, and “smaller than”

relation, R<

Specify rule in English: “a is related to b, if a is smaller

than b”

List all pairs that are related

1 is smaller than 2, 1 is smaller than 3, 2 is smaller than 3.

(1,2),(1,3),(2,3) are all ordered pairs of elements

that are related under R<

i.e., R<={(1,2), (1,3),(2,3)}

Formal definition of binary relation

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For domain S, the set of all possible ordered pairs of

elements from S is the Cartesian product, S x S.

Def: a binary relation R defined on domain S is a

subset of S x S

For example: S={1,2,3}, below are relations on S

R1={(1,2)}

R2={}, no number is related to another number

2}ba and S b and |),{(3 SabaR

Formal definition of binary

relation(cont’d)

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Sometimes relation is between two different sets

“goes to college at” relation is defined from the set of people,

to the set of colleges

Given two sets S and T, a binary relation from S to T is a

subset of SxT.

S is called domain of the relation

T is called codomain of the relation

We focus on binary relation with same domain and

codomain for now.

Domain can be infinite set

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Domain: Z

R: {(a, b) is an element of Z x Z : (a - b) is even}

Given any pair of integers a, b, we can test if they are related under R by checking if a-b is even

e.g., as 5-3=2 is even, 5 is related to 3, or

e.g., as 5-4 is odd,

R)3,5(

R)4,5(

Example

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For the following relation defined on set {1,2,3,4,5,6},

write set enumeration of the relation, and draw a graph

representation:

Rd: “is divisible by”: e.g., 6 is divisible by 2

Rd = {(1,1), (2,1), (3,1), (4,1), (5,1), (6,1),

(4,2), (6,2), (6,3)}

Some exercises

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For each of following relations defined on set {1,2,3,4,5,6},

write set enumeration of the relation, and draw a graph

representation:

R≤: “smaller or equal to”

Ra: “adds up to 6”, e.g., (3,3), (1,5) …

Relationships have properties

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Properties of relations: Reflexive, irreflexive

Symmetric, Anti-symmetric

Transitive

We will introduce the definition of each property and learn to test if a relation has the above properties

Some are immortal.

Primer about negation

Let’s look at a statement that asserts something

about all human being:

All human beings are mortal. (a)

The opposite of statement:

All human beings are immortal.

The negation of statement (˥a):

It’s not true that “all human beings are mortal”

i.e., Some human beings are not mortal.

All are immortal. All are mortal.

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Reflexive Property

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Consider “is the same age as” relation defined on the

set of all people

Does this relation “go back to itself”, i.e., is

everybody related to himself or herself?

◦ Tom is the same age as Tom

◦ Carol is the same age as Carol

◦ Sally is the same age as Sally

For any person, he/she is the same age as himself or herself.

The relation “is the same age as” is reflexive

Reflexive Property

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Def: A relation is reflexive if every element in the domain is related to itself If R is reflexive, there is a loop on every

node in its graph

A relation is not reflexive if there is some element in the domain that is not related to itself As long as you find one element in the

domain that is not related to itself, the relation is not reflexive

Not reflexive since e does not go back to e

Try this mathematical one

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Domain is Z

R={(x, y) | x,y Z, and (x + y) is an even number}

Is R reflexive?

Is any number in Z related to itself under R ?

Try a few numbers, 1, 2, 3, …

For any numbers in Z ?

Yes, since a number added to itself is always even (since 2 will

be a factor), so R is reflexive

Another example

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Domain: R (the set of all real numbers)

Relation: “is larger than”

Try a few examples:

Pick a value 5 and ask “Is 5 larger than 5” ?

No, i.e., 5 is not related to itself

Therefore, this relation is NOT Reflexive

Actually, no real number is larger than itself

No element in R is related to itself, irreflexive relation

Irreflexive Relation

For some relations, no element in the domain is

related to itself.

◦ “greater than” relation defined on R (set of all real

numbers)

1 is not related to itself under this relation, neither is 2 and 3 related

to itself, …

◦ “is older than” relation defined on a set of people

Def: a relation R on domain A is irreflexive if every

element in A is not related to itself

An irreflexive relation’s graph has no self-loop

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For all relations

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irreflexive Reflexive

Not reflexive, not irreflexive

All relations

Every element is related to itself

No element is related to itself

Some element is related to itself, some element is not related to itself

A relation cannot be both reflexive and irreflexive.

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reflexive? irreflexive ? Neither?

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Each of following relations is defined on set {1,2,3,4,5,6},

R≤: “smaller or equal to”

Reflexive, as every number is equal to itself

Ra: “adds up to 6”, e.g., (3,3), (1,5) …

Neither reflexive (as 1 is not related to itself), or irreflexive (as 3 is

related to itself)

R={(1,2),(3,4), (1,1)}

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odd} is :),{( 2 baZZbaR

Symmetric Property

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some relations are mutual, i.e., works both ways, we call them symmetric

E.g., “has the same hair color as” relation among a set of people Pick any two people, say A and B

If A has the same hair color as B, then of course B has the same hair color as A

Thus it is symmetric

Other examples: “is a friend of”, “is the same age as”, “goes to same college as”

In the graphs of symmetric relations, arcs go both ways (with two arrows)

Exercise: symmetric or not

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Domain: {1, 2, 3 ,4}

Relation={(1, 2), (1, 3), (4, 4), (4, 5), (3, 1), (5,4), (2, 1)}

Yes, it is symmetric since

(1,2) and (2,1)

(1,3) and (3,1)

(4,5) and (5,4)

Domain: Z (the set of integers)

Relation: add up to an even number

A relation that is not symmetric

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“is older than” relation

If Sally is older than Tom, then Tom cannot be older than Sally

We found a pair Sally and Tom that relate in one direction, but not the other

Therefore, this relation is not symmetric.

Actually, for “is older than” relation, it never works both way

For any two people, A and B, if A “is older than B”, then B is not older than A.

Anti-symmetric Property

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Some relations never go both way ◦ E.g. “is older than” relation among set of people ◦ For any two persons, A and B, if A is older than B, then B is

not older than A ◦ i.e., the relation never goes two ways

Such relations are called anti-symmetric relations

In the graph, anti-symmetric relations do not have two-way arcs.

Formal Definition of Symmetric

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RabRbaAba ),( then,),( if ,, allfor if

symmetric is A domain on defined Rrelation A

To decide whether a given relation is symmetric, you check whether underlined statement is true

◦ Underlined statement claim something for “all” a and b…

◦ If you find one pair of a and b that makes it false, then the whole statement is false

◦ i.e., if you find a and b, such that (a,b)∈R, and (b,a)∉R, then R is not symmetric

Formal Definition of Anti-symmetric

RabRbabaAba ),( then ),( and if ,, allfor if

symmetric-anti isA domain on defined Rrelation A

To decide whether a given relation is symmetric, you check whether underlined statement is true

◦ Underlined statement claims something for “all” a and b…

◦ If you find one pair of a and b that makes it false, then it is false

◦ If you find a, b, a≠b, (a,b)∈R and (a,b)∈R, then R is not anti-symmetric

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Symmetric & Anti-Symmetric

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Anti-symmetric Symmetric

Not symmetric, not anti-symmetric

All relations

“Know the birthday of”

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Defined among our class of students

For some pair of students, one knows the birthday of the other, but not vice versa

So the relation is not symmetric.

Some pair of people know each other’s birthday

So this relation is not anti-symmetric.

Therefore this relation is neither symmetric nor anti-symmetric

Exercises: Symmetric? Anti-symmetric ?

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For each of following relations defined on set {1,2,3,4,5,6}

R={(1,2),(3,4), (1,1),(2,1),(4,3)}

R={(1,2),(3,4),(1,1),(4,3)}

R≤: “smaller or equal to”

Rd: “divides”: e.g., 6 divides 2

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odd} is :),{( 2 baZZbaR

Recall

Reflexive, irreflexive property

Concerned about whether each object is related to itself or not

Symmetric, anti-symmetric property

Concerned about each pair of objects that are related in one direction, whether they are related in another direction too.

Next, transitive property

Concerned about every set of three objects …

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Transitive: an introduction

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You are assigned a job to draw graph that represents “is older than” relation defined on our class ◦ You can only ask questions such as “Is Alice older than Bob?”

Suppose you already find out: ◦ Alice is older than bob

◦ Bob is older than Cathy

◦ Do you need to ask “Is Alice older than Cathy?” No ! Alice for sure is older than Cathy.

For any three people, a, b and c, if a is older than b, b is older than c, then fore sure, a is older than c.

Such property of this relation is called transitive.

Is this relation transitive ?

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“is taking the same class as” relation on a set of students

Suppose three students, Bob, Katie, and Alex, Bob is taking the same class as Katie

Katie is taking the same class as Alex

And now consider: Is Bob is taking the same class as Alex ?

Many cases: no Bob takes 1400 with Katie, and Katie takes history with

Alex, while Bob and Alex has no classes in common.

Therefore this relation is not transitive

Transitive Property

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A relation R is transitive if for any three elements in the domain, a, b, and c, knowing that a is related to b, and b is related to c would allow us to infer that a is related to c.

E.g. “is older than”, “is same age as” is transitive

a

b

c In graph of a transitive relation: if there is two-hop paths from a to c, then there is one-hop path from a to c.

Not Transitive

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A relation R is not transitive if there exists three elements in the domain, a, b, and c, and a is related to b, b is related to c, but a is not related to c.

E.g. “is taking same class as”, “know birthday of”

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a

b

c

In graph: there is two-hop paths from a to c, but there is not a one-hop path from a to c.

Formal Definition of Transitive

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Rca

RcbRba

),( then

,),( and ),( if

A, cb,a,any for

if e, transitivisA domain on RRelation

Transitive Not transitive

All relations

Exercises: Transitive or not ?

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R≤: “smaller or equal to” defined on set {1,2,3,4,5,6}

For three numbers a, b, c from {1,2,3,4,5,6}

Would knowing that a≤b, and b≤c, allows me to conclude that a ≤c

?

Yes !

It’s transitive !

Let’s check it’s graph …

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Exercises: Transitive or not ?

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Are the following relations defined on set {1,2,3,4,5,6}

transitive ?

Rd: “is divisible by”: e.g., 6 is divisible by 2

Ra: “adds up to 6”, e.g., (3,3), (1,5) …

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Example

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What properties does relation R has ?

odd} is :),{( 2 baZZbaR

Application: Partial Ordering A relation R on a set S that is reflexive, anti-symmetric,

and transitive is called a partial ordering on S. ◦ e.g. “less than or equal to”

◦ e.g., “is a subset of”,

◦ e.g., “is prerequisite of”

◦ If (a,b) is related under R, we call a is predecessor of b, and b is a successor of a.

If furthermore, any two elements in domain is related (in one direction only), then it’s a total ordering ◦ E.g. less than or equal to

◦ Is a subset of: is not a total ordering

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Topological Sorting

Topological sorting: given a partial ordering on S,

order elements in S such that all predecessors appear

before their successors.

e.g., Determine the order of taking courses based on

prerequisite relation

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The idea of algorithm

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Input: a list of ordered pairs each describing prerequisite

requirement

e.g., (CS1, CS2): one needs to take CS1 before taking CS2

E.g. (CS2, Data Structure) …

Output: an ordering of the courses, such that if (c1, c2) is in

the prerequisite relation, then c1 appears before c2 in the

ordering

Algorithm input Output

Design the algorithm

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an algorithm: an effective method for solving a problem using a finite sequence of instructions.

Step by step procedure to generate the output based on the input… ◦ Work for different possible input

◦ One algorithm should work for computer science student, biology student, physics student

E.g. multiple digits addition multiplication

Finding the minimal numbers

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Problem setting

Input: a set of numbers n1,n2,…,nk

Output: the smallest number in the set

How would you do it?

How to describe your approach so that other people

(like programmers) can understand it ?

Finding the minimal numbers

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Problem setting ◦ Input: a list of numbers n1,n2,…,nk ◦ Output: the smallest number in the set

Algorithm_Finding_Minimal 1. Set current minimal value to the first number in the set

2. Compare the next number from the list with the minimal value

3. if the number is smaller than the current minimal value

then

set the minimal value to the number

endif

4. Repeat step 2,3 until reaching the last number in the list

5. Return the current minimal value

Functions

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Functions are everywhere

A function is a way of transforming one set of things

(usually numbers) into another set of things (also usually

numbers).

For example:

Fahrenheit to Celsius Conversion (link)

[°C] = ([°F] − 32) × 5⁄9

Closed formula of a sequence: maps position to value (link)

an=100*n+1, bn =2n

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Components of a function

Name, typically a letter like f, g, h, …

Domain, a set of values

Codomain, a set of values

Rule: maps values in the domain to values in the

codomain

For every value in the domain, the rule maps it into a single

value in the codomain

e.g. , f: R R, f(x)=2x+1

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Function f : S T

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Function f maps values in set S (domain) to values in set

T (codomain)

Domain S Codomain T

t s

f(s)=t f maps s to t t is called the image of s. s is called the pre-image of t.

f

Useful analogy: elements in S: pigeons elements in T: holes f(s)=t: pigeon s flies into hole t

Using mathematic formula

For functions of numbers, the mapping can be specified using formula

f(a) = a + 4, “f of a equals a plus 4”

g(b) = b * b + 2, “g of b equals b times b plus 2”

h(c) = 5, “h of c equals 5”

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Definition of function as relation

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Function f: A →B is a relation with domain A and

codomain B, and for every , there is exactly one

element for which , we write it as f(x)=y.

i.e., a function is a relation where every element in the

domain, say x, is related to exactly one element in the

codomain, say y.

Ax

By fyx ),(

Useful analogy: elements in S: pigeons elements in T: holes f(s)=t: pigeon s flies into hole t every pigeon goes to one hole

Ok, begin the same way (take values from the domain

and put them in the formula)

Choose 0. f(x) = 5 … it’s in the Codomain

Choose 1. f(1) = 6 … it’s in the Codomain

Choose -1. f(-1) = 4 ... it’s in the Codomain

But we can’t do this forever

Hand-waving argument

“Regardless of what integer I take from the domain, I can

add 5 to that number and still have a value in the codomain.”

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Domain = Z (all integers) Codomain = Z (all integers)

f(x) = x + 5

Dealing with infinite sets

If the domain is infinite, you can’t try all values in the

domain

So you need to look for values that might not

work and try those.

If you can’t find any domain values that don’t work, can

you make an argument that all the domain

values do work ?

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Choose some values

Choose 0: f(0) = -1 … -1 is not in the codomain, it doesn’t

work

so it’s not a function

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Domain = N (natural numbers, i.e., 0,1,2,…)

Codomain = N

f(x) = x-1

Functions with multiple variables

Example:

f(x,y)=x-y, where x takes integer value, and y takes integer value

f maps an ordered pair of integers, i.e., x and y, to their difference (x-y), which is also an integer

What’s the domain ?

The set of ordered pairs of integers …

In mathematical notation: , the Cartesian product of Z

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ZZ

Outline

Definition of function

Property of functions

◦ one-to-one

◦ onto

◦ Pigeonhole principle

Inverse function

Function composition

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Properties of Functions

Two interesting properties of functions

one-to-one

onto

Bijective: one-to-one and onto

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One-to-one function

function f: S→T is one-to-one, if no two different

values in the domain are mapped to the same value

in codomain.

for two elements , if s1≠s2, then f(s1)≠f(s2)

Equivalently, for two elements , if f(s1)=f(s2), then

s1=s2

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Sss 21,

Sss 21,

Useful analogy: elements in S: pigeons elements in T: holes f(s)=t: pigeon s flies into hole t every pigeon goes to one hole one-to-one function: no two pigeons go to same hole

Example

So we must ask if it is a function? ◦ Choose 1: f(1) = 2

◦ Choose 2: f(2) = 3

◦ Choose 3: f(3) = 4

◦ So it is a function.

Is it injective (one-to-one)? Well ◦ we only reached value 2 by using x = 1.

◦ we only reached value 3 by using x = 2.

◦ we only reached value 4 by using x = 3.

So it is one-to-one

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Domain = {1, 2, 3} Codomain = {1, 2, 3, 4} f(x) = x + 1

Is it a function?

f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2)=4

So it is a function

Is it one-to-one?

No. We can reach the value 4 in two ways.

f(-2)=4 and f(2)=4

For any injective function f: S→T, where S and T are finite,

what kind of relation hold between |T|, |S| ?

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Domain = {-2, -1, 0, 1, 2} Codomain = {0, 1, 2, 3, 4, 5, 6} f(x) = x2

Pigeonhole Theorem

Consider function f: S→T, where S, T are finite

sets

If f is one-to-one, then |S|≤|T|

If |S|>|T|, then f is not injective.

at least two diff. values in S are mapped to same value in T

Pigeonhole Theorem

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Domain S Codomain T

t s

f

Dealing with infinite sets

Is the function injective (one-to-one) ?

What does this function look like ?

One can use Excel to plot a function (link)

A function is injective if its graph is never intersected by a

horizontal line more than once.

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f: R→R with the rule f(x)=x2+4x+1

Onto Functions

A function is onto if every value in the codomain is the image of some value in the domain (i.e., every value in the codomain is taken) f: S→T is onto if for any element t in T, there exists an element

s in S, such that f(s)=t

Let Range of f, ran(f), be the set of all values that f can take: For onto function f, ran(f)=T

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}:)({)( Sxxffran

Surjective function example

First you have to figure out if it is a function or not. ◦ Choose 1: f(1) = 11

◦ Choose 2: f(2) = 12

◦ Choose 3: f(3) = 13

◦ Choose 4: f(4) = 14

◦ So it is a function

Is it onto? ◦ Yes. Because we covered all of the Codomain values, i.e.,

every value of codomain is an image of some values in the domain.

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Domain = {1, 2, 3, 4} Codomain = {11, 12, 13, 14} f(x) = x + 10

Determine whether it is a function

Choose 1: f(1) = 0

Choose 2: f(2) = 1

Choose 3: f(3) = 2

So it is a function.

Is it onto?

No, we never arrived at the value 3 which is in the Codomain

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Domain = {1, 2, 3} Codomain = {0, 1, 2, 3} f(x) = x -1

Property of onto function

Consider function f: S→T, with S, T finite

If f is onto, then |S|≥|T|

If |S|<|T|, then f is not onto

There is some element in T that is not mapped to

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Domain S Codomain T

t s

f

Useful analogy: elements in S: pigeons elements in T: holes f(s)=t: pigeon s flies into hole t every pigeon goes to one hole onto function: every hole is occupied by some pigeon(s)

Dealing with infinite sets*

Is it a function?

Is it an onto function ?

For every

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f: R→R with the rule f(x)=x2+1

Rs some exists thereR, t

?t 1sf(s)such that 2

Bijection

A function that is both onto and one-to-one is called a

bijection, or we say the function is bijective.

Consider function f: S→T, with S, T finite

If f is bijective (one-to-one and onto), then |S|=|T|

as f is one-to-one, we have |S|≤|T|

as f is onto, we have |S|≥|T|

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Useful analogy: elements in S: pigeons elements in T: holes f(s)=t: pigeon s flies into hole t every pigeon goes to one hole bijective function: every hole is occupied by exactly one pigeon

Invertible functions

Give a function f: S→T, what happens if we “invert” the mapping, and get a new relation ? Make T the new domain

Make S the new codomain

If is mapped to , we now map t to s

Do we get a function this way ? i.e., is any value in the new domain (T) being mapped to

one and only one value in the new domain S ?

If we get a function after we insert the mapping, the f is called invertible.

Ss Tt

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Formal definition of inverse

Function f: A→B is invertible if there is a function f-1:B

→A such that f(x)=y if and only if f-1(y)=x.

f-1: read as “f inverse”

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What kind of function is invertible? Every value in the codomain has values in the domain

mapped to it The function is onto.

<draw a diagram showing a function that is not onto, what happens when reverse mapping…>

Every value in the codomain has only one value in the domain mapped to it. i.e. the function is one-to-one

<draw a diagram of a function that is not one-to-one, what happens if reverse mapping?>

A function is invertible if it’s bijective.

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Finding inverse function

To find inverse of function f

First check if f is invertible (i.e., bijective)

Make the old codomain the new domain

Make the old domain the new codomain

Reverse the mapping

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Reverse the mapping

If the mapping is given by the set of ordered pairs

Just reverse the first- and second- components of each pair

If the function is given by a diagram

Reverse the directions of the arrows

If the function is given by a formula, f(x)

Solve the formula for x, i.e., express x in terms of f(x)

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Inverse Example, f-1

New domain = {4, 8, 12, 16}

New Codomain = {2, 4, 6, 8}

Original mapping maps x to y=f(x)=2x

Reverse mapping map y to x, i.e., given y, what’s the x ?

(express x in terms of y)

Solve y=2x for x, we get x=y/2.

Inverse function is

f-1: {4,8,12,16}→{2,4,6,8}, f-1(y)=y/2

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Domain = {2, 4, 6, 8} Codomain = {4, 8, 12, 16} f(x) = 2x

Examples

Are the following functions invertible ? Find inverse for

those invertible.

f: R→R, with f(x)=3x+6

f:R →R, with f(x)=x2

g:Z →Z, with the rule:

0z if ,12)(

0z if ,2)(

zzg

zzg

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Outline

Definition of function

Property of functions

◦ Onto

◦ One-to-one

◦ Pigeonhole principle

◦ Inverse function

Function composition

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Function Composition

We can chain functions together – this is called

composition (apply the mappings subsequently)

f ◦ g (reads “f composed with g”) defined as f ◦ g

(x)=f(g(x)) (note apply g first, then f)

First apply mapping of g, then apply mapping of f

CSRU1400 Spring 2008 Ellen Zhang 78

g(x) x

f(g(x))

g f f ◦ g

78

Function Composition Example

Example: for f, g with domain/codomain of R

f(x) = x+5, g(x) = 2x + 3

f ◦ g (x)=f(g(x))

=f(2x+3)

=(2x+3) + 5

=2x + 8

g◦ f (x)=g(f(x))

=g(x+5)

= 2(x+5) + 3

= 2x + 13

CSRU1400 Spring 2008 Ellen Zhang 79 79

What is f ◦ g ?

What is g ◦ f?

What is f ◦ f?

What is g ◦ g ?

What is f ◦ g for g(2)?

80

Assume we have two functions with Domains and Codomains over all integers

f(x) = 3x – 2

g(x) = x * x

f(x*x) = 3(x*x)-2 = 3x2-2

g(3x-2) = (3x-2)*(3x-2)=9x2-12x+4

f(3x-2) = 3(3x-2)-2 = 9x-8

g(x*x) = (x*x) * (x*x) = x4

f(g(2)) = f(4) = 3(4) – 2 = 10

Function Compositions

81

Assume we have two functions with Domains and Codomains over all real numbers …

f(x) = 3x – 2

g(x) = x3

h(x) = x/4

What is f-1?

What is f ◦ f-1?

What is f -1◦ f?

What is f ◦ g ◦ f-1 ?

What is f ◦ f-1 ◦ g?